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The polynomial moment pairing
Definition
Fix moment data as in Positive-measure Gaussian quadrature moment data. For an integer with and , let with its coordinatewise real vector-space operations. For , let . This is the evaluation of the zero polynomial when , and otherwise of the formal polynomial obtained by deleting trailing zero coefficients from (Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials). Define the polynomial moment pairing on by This is a candidate real inner product in the sense of Real and complex inner product spaces, with the inner product linear in the first argument; its well-definedness and positive definiteness under the stated support and moment hypotheses are proved in The polynomial moment pairing is a real inner product ↗.
Depends on
Used by
- Weighted orthogonal-polynomial Galerkin spaces and spectral scale Definition
- Monic orthogonal polynomials satisfy a three-term recurrence Lemma
- Gaussian quadrature has the Hermite mean-value remainder Theorem
- Gaussian quadrature is exact through degree 2m−1 and has positive weights Theorem
- Gaussian weights equal the total mass times a Jacobi eigenvector's first-coordinate square Theorem
- Monic orthogonal polynomials exist uniquely through the quadrature order Theorem
- The polynomial moment pairing is a real inner product Theorem
Dependency tree · two levels
14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John A. Gubner, Gaussian Quadrature and the Eigenvalue Problem, §§2–5, 2020 (standard reference, not scraped)
- MATH 373 Lecture Notes, Lecture 14: Gaussian Quadrature, §§10.2–10.5 (standard reference, not scraped)
- Walter Gautschi, The Interplay Between Classical Analysis and (Numerical) Linear Algebra, §2.1 (standard reference, not scraped)